First item - each airline dispatches its own flights, regardless where in the world they are - thats a requirement of positive operational control. AA dispatches all their ops from the AA Flight Academy in DFW - American Eagle is downstairs in their own SOC. UAL is at WHQ (World Headquarters) in suburban Elk Grove Village outside Chicago, and so on.
Pre-9/11 most airlines, except AA and AE, permitted offline dispatchers to ride cockpit jumpseat. Pre 9/11, when I dispatched for a regional partner of UAL (and later for AE), I rode nearly everyone's flight decks as a commuter out of DFW. AA and AE do not permit offline dispatchers to ride, pre-9/11 or post-9/11, even in the back (which is complete B.S., but I digress).
Post 9/11, and I dont think this is much of a secret anymore, no offliners can ride flight deck, but may ride in the back, space and captain permitting. A dispatcher can usually ride his carriers jumpseat, plus those other carriers his airline code shares with, if any, but thats it. I hear high-grade rumors about a new program to open it back up for offliners, but I have heard of no details.
And as for BA - they dont use dispatchers, few other countries (I think Canada is the only other one - but I'm probably wrong on that) have the requirement for licensed dispatchers to share in the exercise of positive operational control. BA has flight ops officers, which do nothing but generate minimum-time-track flight plans and file flight plans - their FOOs have no right (or legally-mandated requirement) for flight following, or exercising other operational control functions.
Several years before 9/11 the FAA Office of General Counsel issued an interpretation of 121.547 about the question of pilots/dispatchers of FAR 129 carriers (foreign air carriers) and their ability to ride 121 flight deck jumpseats - they may not, since their carriers are not "certificate holders" under FAR 121.
